The idea that any type of choice is possible, even those of the compatibilist sort, can be compatible with determinism or randomness is beyond anything I can imagine. I think concepts such as compatibilism exist only because people feel a need to defend some type of free will. To do it they must make up new definitions which are not what people have traditionally meant by free will.
I am not saying that tradition is always right or that we can never have new understandings of terms, but seriously, I think it leads to a lot of confusion when we use a word to mean something different than most people understand it. Sort of the way the words "gay" and "ass" have completely lost their original meanings. Because of this I tend to stick with older definitions of free will that are more clearly incompatible with determinism.
Compatibilists are usually referring to the physical freedom to do what we want. This is also not our choice because we can be stopped by all sorts of limitations we have no control over.